p = 0.50 Site percolation Square lattice 400 x 400 Three largest clusters are coloured green/blue/yellow p = 0.50

p = 0.55

p = 0.58

p = 0.59

p = 0.60

p = 0.65

Fraction of sites on the largest cluster

S = mean size of finite clusters Estimate in Bunde and Havlin: pc = 0.5927

Cluster generation via Leath method (epidemic spreading)

M ~ rdf r space dimension d = 2 fractal dimension df r At pc the infinite cluster has a fractal dimension df < 2 r

Estimate of fractal dimension of percolation clusters generated by Leath method at p = 0.59 Exact answer : df = 91/48 = 1.896

p = 0.55  For p < pc correlation length  = mean distance between points on the same finite cluster p = 0.55 

 For p > pc can still define correlation length = mean distance between points on the same finite cluster. This is typical size of holes in infinite cluster. The infinite cluster is uniform above this length scale 

p = 0.65 Minimum path length from centre. red = short green = long almost circular contours  uniform medium p = 0.65

p = 0.59 Minimum path length from centre. red = short green = long irregular contours  poorly connected medium fractal p = 0.59

p = 0.72 Diffusion through the infinite cluster Concentration red = 1.0, green = 0.0 Flux red = high, blue = low

Diffusion through the infinite cluster close to percolation Concentration red = 1.0, green = 0.0 Flux red = high, blue = low

lmin ~ rdmin L = 100 Shortest path across a cluster close to pc

lmin ~ Ldmin Shortest path across lattice of size L Estimate in Bunde and Havlin book = 1.13

Testing the scaling hypothesis for cluster size distribution.